THE TRACY-WIDOM LAW FOR THE LARGEST EIGENVALUE OF F TYPE MATRICES

Let ${A_P} = \frac{{YY*}}{m}$ and ${B_p} = \frac{{XX*}}{n}$ be two independent random matrices where X = (Xij)p.n and Y = (Yij)P.m respectively consist of real (or complex) independent random variables with ..Xij = ..Yij = 0, ..|Xij|² = ..|Yij|² = 1. Denote by .. the largest root of the determinantal equation det(.Ap . Bp) = 0. We establish the Tracy-Widom type universality for .. under some moment conditions on Xij and Yij when p/m and p/n approach positive constants as p . ..